Fire effect looks different when renderd
First off I would disagree that the fire looks "better" in the viewport (but that is the very nature of using comparative and superlative adjectives to ask questions: at the end they are just opinions... Impossible to answer objectively...)

If the issue is how bright the fire is, there are many ways to control it.

The most obvious way is by increasing the intensity of the emission for the smoke domain.



It is very easy to push the values to the point where the brightest (hottest) part of the fire become white.

The issue becomes that the intensity of the emitter prevents us very quickly to deal with the color saturation in a way that resembles the images of fire that we are used to.

A different approach would be to change the intensity along with the way the values of the scene are mapped to create an image, so....

Welcome to the wonderful world of color management!

In this brave, and mostly unexplored world, you can have a lot more control on the intensity, and most importantly, how those values become the final image.

Given that you are exploring a version of blender that includes the filmic blender, you can find new ways to have intense values without having to loose detail in the bright areas.